/*      rpoly.cpp -- Jenkins-Traub real polynomial root finder.
 *
 *      (C) 2000, C. Bond.  All rights reserved.
 *
 *      Translation of TOMS493 from FORTRAN to C. This
 *      implementation of Jenkins-Traub partially adapts
 *      the original code to a C environment by restruction
 *      many of the 'goto' controls to better fit a block
 *      structured form. It also eliminates the global memory
 *      allocation in favor of local, dynamic memory management.
 *
 *      The calling conventions are slightly modified to return
 *      the number of roots found as the function value.
 *
 *      INPUT:
 *      op - double precision vector of coefficients in order of
 *              decreasing powers.
 *      degree - integer degree of polynomial
 *
 *      OUTPUT:
 *      zeror,zeroi - output double precision vectors of the
 *              real and imaginary parts of the zeros.
 *
 *      RETURN:
 *      returnval:   -1 if leading coefficient is zero, otherwise
 *                  number of roots found. 
 */

#ifndef RPOLY_H_
#define RPOLY_H_

#include "math.h"

class RootFinder
{
public:
	int rpoly(const double *op, int degree, double *zeror, double *zeroi) 
	{
		double t,aa,bb,cc,factor,rot;
		double lo,max,min,xx,yy,cosr,sinr,xxx,x,sc,bnd;
		double xm,ff,df,dx,infin,smalno,base;
		int cnt,nz,i,j,jj,l,nm1,zerok;
		/*  The following statements set machine constants. */
		base = 2.0;
		eta = 2.22e-16;
		infin = 3.4e38;
		smalno = 1.2e-38;

		are = eta;
		mre = eta;
		lo = smalno/eta;
		/*  Initialization of constants for shift rotation. */        
		xx = sqrt(0.5);
		yy = -xx;
		rot = 94.0;
		rot *= 0.017453293;
		cosr = cos(rot);
		sinr = sin(rot);
		n = degree;
		/*  Algorithm fails of the leading coefficient is zero. */
		if (op[0] == 0.0) return -1;
		/*  Remove the zeros at the origin, if any. */
		while (op[n] == 0.0) {
			j = degree - n;
			zeror[j] = 0.0;
			zeroi[j] = 0.0;
			n--;
		}
		if (n < 1) return -1;
		/*
		 *  Allocate memory here
		 */
                double temp[7];
                //		temp = new double [degree+1];
                double pt[7];
                //		pt = new double [degree+1];
                //		svk = new double [degree+1];
		/*  Make a copy of the coefficients. */
		for (i=0;i<=n;i++)
			p[i] = op[i];

		/*  Start the algorithm for one zero. */
_40:        
		if (n == 1) {
			zeror[degree-1] = -p[1]/p[0];
			zeroi[degree-1] = 0.0;
			n -= 1;
			goto _99;
		}
		/*  Calculate the final zero or pair of zeros. */
		if (n == 2) {
			quad(p[0],p[1],p[2],&zeror[degree-2],&zeroi[degree-2],
					&zeror[degree-1],&zeroi[degree-1]);
			n -= 2;
			goto _99;
		}
		/*  Find largest and smallest moduli of coefficients. */
		max = 0.0;
		min = infin;
		for (i=0;i<=n;i++) {
			x = fabs(p[i]);
			if (x > max) max = x;
			if (x != 0.0 && x < min) min = x;
		}
		/*  Scale if there are large or very small coefficients.
		 *  Computes a scale factor to multiply the coefficients of the
		 *  polynomial. The scaling si done to avoid overflow and to
		 *  avoid undetected underflow interfering with the convergence
		 *  criterion. The factor is a power of the base.
		 */
		sc = lo/min;
		if (sc > 1.0 && infin/sc < max) goto _110;
		if (sc <= 1.0) {
			if (max < 10.0) goto _110;
			if (sc == 0.0)
				sc = smalno;
		}
		l = (int)(log(sc)/log(base) + 0.5);
		factor = pow(base*1.0,l);
		if (factor != 1.0) {
			for (i=0;i<=n;i++) 
				p[i] = factor*p[i];     /* Scale polynomial. */
		}

_110:
		/*  Compute lower bound on moduli of roots. */
		for (i=0;i<=n;i++) {
			pt[i] = (fabs(p[i]));
		}
		pt[n] = - pt[n];
		/*  Compute upper estimate of bound. */
		x = exp((log(-pt[n])-log(pt[0])) / (double)n);
		/*  If Newton step at the origin is better, use it. */        
		if (pt[n-1] != 0.0) {
			xm = -pt[n]/pt[n-1];
			if (xm < x)  x = xm;
		}
		/*  Chop the interval (0,x) until ff <= 0 */
		while (1) {
			xm = x*0.1;
			ff = pt[0];
			for (i=1;i<=n;i++) 
				ff = ff*xm + pt[i];
			if (ff <= 0.0) break;
			x = xm;
		}
		dx = x;
		/*  Do Newton interation until x converges to two 
		 *  decimal places. 
		 */
		while (fabs(dx/x) > 0.005) {
			ff = pt[0];
			df = ff;
			for (i=1;i<n;i++) { 
				ff = ff*x + pt[i];
				df = df*x + ff;
			}
			ff = ff*x + pt[n];
			dx = ff/df;
			x -= dx;
		}
		bnd = x;
		/*  Compute the derivative as the initial k polynomial
		 *  and do 5 steps with no shift.
		 */
		nm1 = n - 1;
		for (i=1;i<n;i++)
			k[i] = (double)(n-i)*p[i]/(double)n;
		k[0] = p[0];
		aa = p[n];
		bb = p[n-1];
		zerok = (k[n-1] == 0);
		for(jj=0;jj<5;jj++) {
			cc = k[n-1];
			if (!zerok) {
				/*  Use a scaled form of recurrence if value of k at 0 is nonzero. */             
				t = -aa/cc;
				for (i=0;i<nm1;i++) {
					j = n-i-1;
					k[j] = t*k[j-1]+p[j];
				}
				k[0] = p[0];
				zerok = (fabs(k[n-1]) <= fabs(bb)*eta*10.0);
			}
			else {
				/*  Use unscaled form of recurrence. */
				for (i=0;i<nm1;i++) {
					j = n-i-1;
					k[j] = k[j-1];
				}
				k[0] = 0.0;
				zerok = (k[n-1] == 0.0);
			}
		}
		/*  Save k for restarts with new shifts. */
		for (i=0;i<n;i++) 
			temp[i] = k[i];
		/*  Loop to select the quadratic corresponding to each new shift. */
		for (cnt = 0;cnt < 20;cnt++) {
			/*  Quadratic corresponds to a double shift to a            
			 *  non-real point and its complex conjugate. The point
			 *  has modulus bnd and amplitude rotated by 94 degrees
			 *  from the previous shift.
			 */ 
			xxx = cosr*xx - sinr*yy;
			yy = sinr*xx + cosr*yy;
			xx = xxx;
			sr = bnd*xx;
			si = bnd*yy;
			u = -2.0 * sr;
			v = bnd;
			fxshfr(20*(cnt+1),&nz);
			if (nz != 0) {
				/*  The second stage jumps directly to one of the third
				 *  stage iterations and returns here if successful.
				 *  Deflate the polynomial, store the zero or zeros and
				 *  return to the main algorithm.
				 */
				j = degree - n;
				zeror[j] = szr;
				zeroi[j] = szi;
				n -= nz;
				for (i=0;i<=n;i++)
					p[i] = qp[i];
				if (nz != 1) {
					zeror[j+1] = lzr;
					zeroi[j+1] = lzi;
				}
				goto _40;
			}
			/*  If the iteration is unsuccessful another quadratic
			 *  is chosen after restoring k.
			 */
			for (i=0;i<n;i++) {
				k[i] = temp[i];
			}
		} 
		/*  Return with failure if no convergence with 20 shifts. */
_99:
                /*		delete [] svk;
		delete [] qk;
		delete [] k;
		delete [] qp;
		delete [] p;
		delete [] pt;
		delete [] temp;*/

		return degree - n;
	}

protected:

	double p[7],qp[7],k[7],qk[7],svk[7];
	double sr,si,u,v,a,b,c,d,a1,a2;
	double a3,a6,a7,e,f,g,h,szr,szi,lzr,lzi;
	double eta,are,mre;
	int n,nn,nmi,zerok;

	/*  Computes up to L2 fixed shift k-polynomials,
	 *  testing for convergence in the linear or quadratic
	 *  case. Initiates one of the variable shift
	 *  iterations and returns with the number of zeros
	 *  found.
	 */
	void fxshfr(int l2,int *nz)
	{
		double svu,svv,ui,vi,s;
		double betas,betav,oss,ovv,ss,vv,ts,tv;
		double ots,otv,tvv,tss;
		int type, i,j,iflag,vpass,spass,vtry,stry;

		*nz = 0;
		betav = 0.25;
		betas = 0.25;
		oss = sr;
		ovv = v;
		/*  Evaluate polynomial by synthetic division. */
		quadsd(n,&u,&v,p,qp,&a,&b);
		calcsc(&type);
		for (j=0;j<l2;j++) {
			/*  Calculate next k polynomial and estimate v. */
			nextk(&type);
			calcsc(&type);
			newest(type,&ui,&vi);
			vv = vi;
			/*  Estimate s. */
			ss = 0.0;
			if (k[n-1] != 0.0) ss = -p[n]/k[n-1];
			tv = 1.0;
			ts = 1.0;
			if (j == 0 || type == 3) goto _70;
			/*  Compute relative measures of convergence of s and v sequences. */
			if (vv != 0.0) tv = fabs((vv-ovv)/vv);
			if (ss != 0.0) ts = fabs((ss-oss)/ss);
			/*  If decreasing, multiply two most recent convergence measures. */
			tvv = 1.0;
			if (tv < otv) tvv = tv*otv;
			tss = 1.0;
			if (ts < ots) tss = ts*ots;
			/*  Compare with convergence criteria. */
			vpass = (tvv < betav);
			spass = (tss < betas);
			if (!(spass || vpass)) goto _70;
			/*  At least one sequence has passed the convergence test.
			 *  Store variables before iterating.
			 */
			svu = u;
			svv = v;
			for (i=0;i<n;i++) {
				svk[i] = k[i];
			}
			s = ss;
			/*  Choose iteration according to the fastest converging
			 *  sequence.
			 */
			vtry = 0;
			stry = 0;
			if ( (spass && (!vpass)) || tss < tvv) goto _40;
_20:        
			quadit(&ui,&vi,nz);
			if (*nz > 0) return;
			/*  Quadratic iteration has failed. Flag that it has
			 *  been tried and decrease the convergence criterion.
			 */
			vtry = 1;
			betav *= 0.25;
			/*  Try linear iteration if it has not been tried and
			 *  the S sequence is converging.
			 */
			if (stry || !spass) goto _50;
			for (i=0;i<n;i++) {
				k[i] = svk[i];
			}
_40:
			realit(&s,nz,&iflag);
			if (*nz > 0) return;
			/*  Linear iteration has failed. Flag that it has been
			 *  tried and decrease the convergence criterion.
			 */
			stry = 1;
			betas *=0.25;
			if (iflag == 0) goto _50;
			/*  If linear iteration signals an almost double real
			 *  zero attempt quadratic iteration.
			 */
			ui = -(s+s);
			vi = s*s;
			goto _20;
			/*  Restore variables. */
_50:
			u = svu;
			v = svv;
			for (i=0;i<n;i++) {
				k[i] = svk[i];
			}
			/*  Try quadratic iteration if it has not been tried
			 *  and the V sequence is convergin.
			 */
			if (vpass && !vtry) goto _20;
			/*  Recompute QP and scalar values to continue the
			 *  second stage.
			 */
			quadsd(n,&u,&v,p,qp,&a,&b);
			calcsc(&type);
_70:
			ovv = vv;
			oss = ss;
			otv = tv;
			ots = ts;
		}
	}
	/*  Variable-shift k-polynomial iteration for a
	 *  quadratic factor converges only if the zeros are
	 *  equimodular or nearly so.
	 *  uu, vv - coefficients of starting quadratic.
	 *  nz - number of zeros found.
	 */
	void quadit(double *uu,double *vv,int *nz)
	{
		double ui,vi;
		double mp,omp,ee,relstp,t,zm;
		int type,i,j,tried;

		*nz = 0;
		tried = 0;
		u = *uu;
		v = *vv;
		j = 0;
		/*  Main loop. */
_10:    
		quad(1.0,u,v,&szr,&szi,&lzr,&lzi);
		/*  Return if roots of the quadratic are real and not
		 *  close to multiple or nearly equal and of opposite
		 *  sign.
		 */
		if (fabs(fabs(szr)-fabs(lzr)) > 0.01 * fabs(lzr)) return;
		/*  Evaluate polynomial by quadratic synthetic division. */
		quadsd(n,&u,&v,p,qp,&a,&b);
		mp = fabs(a-szr*b) + fabs(szi*b);
		/*  Compute a rigorous bound on the rounding error in
		 *  evaluating p.
		 */
		zm = sqrt(fabs(v));
		ee = 2.0*fabs(qp[0]);
		t = -szr*b;
		for (i=1;i<n;i++) {
			ee = ee*zm + fabs(qp[i]);
		}
		ee = ee*zm + fabs(a+t);
		ee *= (5.0 *mre + 4.0*are);
		ee = ee - (5.0*mre+2.0*are)*(fabs(a+t)+fabs(b)*zm)+2.0*are*fabs(t);
		/*  Iteration has converged sufficiently if the
		 *  polynomial value is less than 20 times this bound.
		 */
		if (mp <= 20.0*ee) {
			*nz = 2;
			return;
		}
		j++;
		/*  Stop iteration after 20 steps. */
		if (j > 20) return;
		if (j < 2) goto _50;
		if (relstp > 0.01 || mp < omp || tried) goto _50;
		/*  A cluster appears to be stalling the convergence.
		 *  Five fixed shift steps are taken with a u,v close
		 *  to the cluster.
		 */
		if (relstp < eta) relstp = eta;
		relstp = sqrt(relstp);
		u = u - u*relstp;
		v = v + v*relstp;
		quadsd(n,&u,&v,p,qp,&a,&b);
		for (i=0;i<5;i++) {
			calcsc(&type);
			nextk(&type);
		}
		tried = 1;
		j = 0;
_50:
		omp = mp;
		/*  Calculate next k polynomial and new u and v. */
		calcsc(&type);
		nextk(&type);
		calcsc(&type);
		newest(type,&ui,&vi);
		/*  If vi is zero the iteration is not converging. */
		if (vi == 0.0) return;
		relstp = fabs((vi-v)/vi);
		u = ui;
		v = vi;
		goto _10;
	}
	/*  Variable-shift H polynomial iteration for a real zero.
	 *  sss - starting iterate
	 *  nz  - number of zeros found
	 *  iflag - flag to indicate a pair of zeros near real axis.
	 */
	void realit(double *sss, int *nz, int *iflag)
	{
		double pv,kv,t,s;
		double ms,mp,omp,ee;
		int i,j;

		*nz = 0;
		s = *sss;
		*iflag = 0;
		j = 0;
		/*  Main loop */
		while (1) {
			pv = p[0];
			/*  Evaluate p at s. */
			qp[0] = pv;
			for (i=1;i<=n;i++) {
				pv = pv*s + p[i];
				qp[i] = pv;
			}
			mp = fabs(pv);
			/*  Compute a rigorous bound on the error in evaluating p. */
			ms = fabs(s);
			ee = (mre/(are+mre))*fabs(qp[0]);
			for (i=1;i<=n;i++) {
				ee = ee*ms + fabs(qp[i]);
			}
			/*  Iteration has converged sufficiently if the polynomial
			 *  value is less than 20 times this bound.
			 */
			if (mp <= 20.0*((are+mre)*ee-mre*mp)) {
				*nz = 1;
				szr = s;
				szi = 0.0;
				return;
			}
			j++;
			/*  Stop iteration after 10 steps. */
			if (j > 10) return;
			if (j < 2) goto _50;
			if (fabs(t) > 0.001*fabs(s-t) || mp < omp) goto _50;
			/*  A cluster of zeros near the real axis has been
			 *  encountered. Return with iflag set to initiate a
			 *  quadratic iteration.
			 */
			*iflag = 1;
			*sss = s;
			return;
			/*  Return if the polynomial value has increased significantly. */
_50:
			omp = mp;
			/*  Compute t, the next polynomial, and the new iterate. */
			kv = k[0];
			qk[0] = kv;
			for (i=1;i<n;i++) {
				kv = kv*s + k[i];
				qk[i] = kv;
			}
			if (fabs(kv) <= fabs(k[n-1])*10.0*eta) {
				/*  Use unscaled form. */
				k[0] = 0.0;
				for (i=1;i<n;i++) {
					k[i] = qk[i-1];
				}
			}
			else {
				/*  Use the scaled form of the recurrence if the value
				 *  of k at s is nonzero.
				 */
				t = -pv/kv;
				k[0] = qp[0];
				for (i=1;i<n;i++) {
					k[i] = t*qk[i-1] + qp[i];
				}
			}
			kv = k[0];
			for (i=1;i<n;i++) {
				kv = kv*s + k[i];
			}
			t = 0.0;
			if (fabs(kv) > fabs(k[n-1]*10.0*eta)) t = -pv/kv;
			s += t;
		}
	}

	/*  This routine calculates scalar quantities used to
	 *  compute the next k polynomial and new estimates of
	 *  the quadratic coefficients.
	 *  type - integer variable set here indicating how the
	 *  calculations are normalized to avoid overflow.
	 */
	void calcsc(int *type)
	{
		/*  Synthetic division of k by the quadratic 1,u,v */    
		quadsd(n-1,&u,&v,k,qk,&c,&d);
		if (fabs(c) > fabs(k[n-1]*100.0*eta)) goto _10;
		if (fabs(d) > fabs(k[n-2]*100.0*eta)) goto _10;
		*type = 3;
		/*  Type=3 indicates the quadratic is almost a factor of k. */
		return;
_10:
		if (fabs(d) < fabs(c)) {
			*type = 1;
			/*  Type=1 indicates that all formulas are divided by c. */   
			e = a/c;
			f = d/c;
			g = u*e;
			h = v*b;
			a3 = a*e + (h/c+g)*b;
			a1 = b - a*(d/c);
			a7 = a + g*d + h*f;
			return;
		}
		*type = 2;
		/*  Type=2 indicates that all formulas are divided by d. */
		e = a/d;
		f = c/d;
		g = u*b;
		h = v*b;
		a3 = (a+g)*e + h*(b/d);
		a1 = b*f-a;
		a7 = (f+u)*a + h;
	}
	/*  Computes the next k polynomials using scalars 
	 *  computed in calcsc.
	 */
	void nextk(int *type)
	{
		double temp;
		int i;

		if (*type == 3) {
			/*  Use unscaled form of the recurrence if type is 3. */
			k[0] = 0.0;
			k[1] = 0.0;
			for (i=2;i<n;i++) {
				k[i] = qk[i-2];
			}
			return;
		}
		temp = a;
		if (*type == 1) temp = b;
		if (fabs(a1) <= fabs(temp)*eta*10.0) {
			/*  If a1 is nearly zero then use a special form of the
			 *  recurrence.
			 */
			k[0] = 0.0;
			k[1] = -a7*qp[0];
			for(i=2;i<n;i++) {
				k[i] = a3*qk[i-2] - a7*qp[i-1];
			}
			return;
		}
		/*  Use scaled form of the recurrence. */
		a7 /= a1;
		a3 /= a1;
		k[0] = qp[0];
		k[1] = qp[1] - a7*qp[0];
		for (i=2;i<n;i++) {
			k[i] = a3*qk[i-2] - a7*qp[i-1] + qp[i];
		}
	}
	/*  Compute new estimates of the quadratic coefficients
	 *  using the scalars computed in calcsc.
	 */
	void newest(int type,double *uu,double *vv)
	{
		double a4,a5,b1,b2,c1,c2,c3,c4,temp;

		/* Use formulas appropriate to setting of type. */
		if (type == 3) {
			/*  If type=3 the quadratic is zeroed. */
			*uu = 0.0;
			*vv = 0.0;
			return;
		}
		if (type == 2) {
			a4 = (a+g)*f + h;
			a5 = (f+u)*c + v*d;
		}
		else {
			a4 = a + u*b +h*f;
			a5 = c + (u+v*f)*d;
		}
		/*  Evaluate new quadratic coefficients. */
		b1 = -k[n-1]/p[n];
		b2 = -(k[n-2]+b1*p[n-1])/p[n];
		c1 = v*b2*a1;
		c2 = b1*a7;
		c3 = b1*b1*a3;
		c4 = c1 - c2 - c3;
		temp = a5 + b1*a4 - c4;
		if (temp == 0.0) {
			*uu = 0.0;
			*vv = 0.0;
			return;
		}
		*uu = u - (u*(c3+c2)+v*(b1*a1+b2*a7))/temp;
		*vv = v*(1.0+c4/temp);
		return;
	}

	/*  Divides p by the quadratic 1,u,v placing the quotient
	 *  in q and the remainder in a,b.
	 */
	void quadsd(int nn,double *u,double *v,double *p,double *q,
			double *a,double *b)
	{
		double c;
		int i;
		*b = p[0];
		q[0] = *b;
		*a = p[1] - (*b)*(*u);
		q[1] = *a;
		for (i=2;i<=nn;i++) {
			c = p[i] - (*a)*(*u) - (*b)*(*v);
			q[i] = c;
			*b = *a;
			*a = c;
		}
	}
	/*  Calculate the zeros of the quadratic a*z^2 + b1*z + c.
	 *  The quadratic formula, modified to avoid overflow, is used 
	 *  to find the larger zero if the zeros are real and both
	 *  are complex. The smaller real zero is found directly from 
	 *  the product of the zeros c/a.
	 */
	void quad(double a,double b1,double c,double *sr,double *si,
			double *lr,double *li)
	{
		double b,d,e;

		if (a == 0.0) {         /* less than two roots */
			if (b1 != 0.0)     
				*sr = -c/b1;
			else 
				*sr = 0.0;
			*lr = 0.0;
			*si = 0.0;
			*li = 0.0;
			return;
		}
		if (c == 0.0) {         /* one real root, one zero root */
			*sr = 0.0;
			*lr = -b1/a;
			*si = 0.0;
			*li = 0.0;
			return;
		}
		/* Compute discriminant avoiding overflow. */
		b = b1/2.0;
		if (fabs(b) < fabs(c)) { 
			if (c < 0.0) 
				e = -a;
			else
				e = a;
			e = b*(b/fabs(c)) - e;
			d = sqrt(fabs(e))*sqrt(fabs(c));
		}
		else {
			e = 1.0 - (a/b)*(c/b);
			d = sqrt(fabs(e))*fabs(b);
		}
		if (e < 0.0) {      /* complex conjugate zeros */
			*sr = -b/a;
			*lr = *sr;
			*si = fabs(d/a);
			*li = -(*si);
		}
		else {
			if (b >= 0.0)   /* real zeros. */
				d = -d;
			*lr = (-b+d)/a;
			*sr = 0.0;
			if (*lr != 0.0) 
				*sr = (c/ *lr)/a;
			*si = 0.0;
			*li = 0.0;
		}
	}
};

#endif


